intro et algo propre. Proof en cours
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Amaury JOLY
@ -1,73 +1,60 @@
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We define $k$ as the id of the round \\
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the $getMax(proves)$ return $MAX({(\_, r) : \exists (\_, PROVE(r)) \in proves})$ \\
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$buffer$ a FIFO list with $buffer[front]$ returning the first element
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We consider a set of processes communicating asynchronously over reliable point-to-point channels. Each process maintains the following shared variables:
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\begin{itemize}
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\item \textbf{received}: the set of messages received (but not yet delivered).
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\item \textbf{delivered}: the set of messages that have been received, ordered, and delivered.
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\item \textbf{prop[$r$][$j$]}: the proposal set of process $j$ at round $r$. It contains the set of messages that process $j$ claims to have received but not yet delivered at round $r$, concatenated with its newly broadcast message.
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\item \textbf{proves}: the current content of the \texttt{DenyList} registry, accessible via the operation \texttt{READ()}. It returns a list of tuples $(j, \texttt{PROVE}(r))$, each indicating that process $j$ has issued a valid \texttt{PROVE} for round $r$.
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\item \textbf{winner$^r$}: the set of processes that have issued a valid \texttt{PROVE} operation for round $r$.
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\item \textbf{RB-cast}: a reliable broadcast primitive that satisfies the properties defined in Section~1.1.2.
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\item \textbf{APPEND$(r)$}, \textbf{PROVE$(r)$}: operations that respectively insert (APPEND) and attest (PROVE) the participation of a process in round $r$ in the DenyList registry.
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\item \textbf{READ()}: retrieves the current local view of valid operations (APPENDs and PROVEs) from the DenyList.
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\item \textbf{ordered$(S)$}: returns a deterministic total order over a set $S$ of messages (e.g., via hash or lexicographic order).
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\end{itemize}
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\begin{algorithm}[H]
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\DontPrintSemicolon
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\SetAlgoLined
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\resetalgline
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\begin{algorithm}
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$rcved = rcved \bigcup \{m\}$ \;
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\textbf{upon} RB\_deliver(PROP, r, S) \textbf{from} j \;
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prop[r][j] = S
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\vspace{1em}
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\textbf{RB-received$(m, S, r_0, j_0)$}
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\begin{algorithmic}[1]
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\State \nextalgline $\textit{received} \gets \textit{received} \cup \{m\}$
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\State \nextalgline $\textit{prop}[r_0][j_0] \gets S$
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\end{algorithmic}
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\caption{\textbf{Upon} RB\_deliver(m)}
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\end{algorithm}
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\vspace{1em}
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\textbf{AB-broadcast$(m, j_0)$}
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\begin{algorithmic}[1]
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\State \nextalgline $\textit{proves} \gets \texttt{READ}()$
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\State \nextalgline $r_0 \gets \max\{r : \exists j,\ (j, \texttt{PROVE}(r)) \in \textit{proves}\} + 1$
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\State \nextalgline $\texttt{RB-cast}(m, (\textit{received} \setminus \textit{delivered}) \cup \{m\}, r_0, j_0)$
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\State \nextalgline \texttt{PROVE}$(r_0)$
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\State \nextalgline \texttt{APPEND}$(r_0)$
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\Repeat
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\State \nextalgline $\textit{proves} \gets \texttt{READ}()$
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\State \nextalgline $r_1 \gets \max\{r : \exists j,\ (j, \texttt{PROVE}(r)) \in \textit{proves}\} - 1$
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\State \nextalgline $\textit{winner}^{r_1} \gets \{j : (j, \texttt{PROVE}(r_1)) \in \textit{proves}\}$
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\State \nextalgline \textbf{wait} $\forall j \in \textit{winner}^{r_1},\ \textit{prop}[r_1][j] \neq \bot$
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\Until{\nextalgline $\forall r_2,\ \exists j_2 \in \textit{winner}^{r_2},\ m \in \textit{prop}[r_2][j_2]$} \nextalgline
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\end{algorithmic}
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\begin{algorithm}[H]
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\DontPrintSemicolon
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\SetAlgoLined
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\KwIn{le message $m$}
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\KwData{rcved = $\emptyset$ \;
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delivered = $\emptyset$ \;
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r = 0 \;}
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\BlankLine
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RB\_cast(m) \;
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$rcved = rcvd \bigcup \{m\}$ \;
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\While{true}{
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$r = r+1$ \;
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$RB\_cast(PROP, r, S)$ \;
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PROVE(r) \;
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APPEND(r) \;
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proves = READ() \;
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$winner^r = \{j : (j, PROVE(r)) \in proves\}$ \;
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\textbf{wait until} $(\forall j \in winner^k: prop[r][j])$ \;
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\If{$\exists j \in winner^k : m \in prop[r][j]$}{
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break \;
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}
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}
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\caption{AB\_Broadcast}
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\end{algorithm}
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\begin{algorithm}[H]
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\DontPrintSemicolon
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\SetAlgoLined
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r\_prev = 0 \;
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\While{true}{
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proves = READ() \;
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$r\_max = MAX(\{r : \exists i, (i, PROVE(r)) \in proves\})$ \;
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\For{$r = r\_prev + 1 \textbf{to} r\_max$}{
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APPEND(r) \;
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proves = READ() \;
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$winner^k = \{j : (j, PROVE(r)) \in proves \}$ \;
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$\textbf{wait until} (\forall j \in winner^k : prop[r][j] \neq \emptyset)$ \;
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$M^r = (\bigcup_{j \in winner^k} prop[r][j]) \setminus delivered$ \;
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\tcc*{we assume $M^r$ as an ordered list s.a. $\forall m_1, m_2, if m_1 < m_2$, $m_1$ appears before $m_2$ in $M^r$}
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\BlankLine
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\ForEach{$m \in M^r$}{
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$delivered = delivered \bigcup \{m\}$ \;
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AB\_deliver(m) \;
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}
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}
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}
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\caption{AB\_Listen}
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\vspace{1em}
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\textbf{AB-listen}
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\begin{algorithmic}[1]
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\While{true}
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\State \nextalgline $\textit{proves} \gets \texttt{READ}()$
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\State \nextalgline $r_1 \gets \max\{r : \exists j,\ (j, \texttt{PROVE}(r)) \in \textit{proves}\} - 1$
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\For{$r_2 \in [r_0, \dots, r_1]$} \nextalgline
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\State \nextalgline \texttt{APPEND}$(r_2)$
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\State \nextalgline $\textit{proves} \gets \texttt{READ}()$
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\State \nextalgline $\textit{winner}^{r_2} \gets \{j : (i, \texttt{PROVE}(r_2)) \in \textit{proves}\}$
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\State \nextalgline \textbf{wait} $\forall j \in \textit{winner}^{r_2},\ \textit{prop}[r_2][j] \neq \bot$
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\State \nextalgline $M^{r_2} \gets \bigcup_{j \in \textit{winner}^{r_2}} \textit{prop}[r_2][j]$
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\ForAll{$m \in \texttt{ordered}(M^{r_2})$} \nextalgline
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\State \nextalgline $\textit{delivered} \gets \textit{delivered} \cup \{m\}$
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\State \nextalgline \texttt{AB-deliver}$(m)$
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\EndFor
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\EndFor
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\EndWhile
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\end{algorithmic}
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\end{algorithm}
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